Temperament mapping matrix: Difference between revisions

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These are dual, in a certain sense, to [[Subgroup Basis Matrices|subgroup basis matrices]], which can be thought of as "co-tempering" vals in the same way that temperament mapping matrices "temper" monzos.

Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix M is said to be a mapping matrix for a temperament T if and only if the right nullspace of M consists of the kernel of T, M is of full row rank, and the rows of M generate a subgroup of the dual group of vals which is [[Saturation|saturated]]. There is generally not a unique matrix M satisfying this definition for arbitrary temperament T, as for any M which is a valid mapping for T, any matrix U∙M where U is unimodular will also be a valid mapping for T. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the quotient group of tempered intervals.

Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix M is said to be a mapping matrix for a temperament T if and only if the right nullspace of M consists of the kernel of T, M is of full row rank, and the rows of M generate a subgroup of the dual group of vals which is [[Saturation|saturated]] (or [[defactored]]). There is generally not a unique matrix M satisfying this definition for arbitrary temperament T, as for any M which is a valid mapping for T, any matrix U∙M where U is unimodular will also be a valid mapping for T. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the quotient group of tempered intervals.

The integer column span of any mapping matrix is the quotient group of T-tempered intervals, also known as the quotient group of '''tempered monzos''' [[Tmonzos_and_Tvals|tmonzos]] for T. The integer row span of any mapping matrix for a temperament T is the subgroup of vals that all support T. Note also that this means that if T is of rank r, then any rank-r matrix in which the rows span the subgroup of vals supporting T will be a valid mapping for T.

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 These are dual, in a certain sense, to [[Subgroup Basis Matrices|subgroup basis matrices]], which can be thought of as "co-tempering" vals in the same way that temperament mapping matrices "temper" monzos.
-Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix M is said to be a mapping matrix for a temperament T if and only if the right nullspace of M consists of the kernel of T, M is of full row rank, and the rows of M generate a subgroup of the dual group of vals which is [[Saturation|saturated]]. There is generally not a unique matrix M satisfying this definition for arbitrary temperament T, as for any M which is a valid mapping for T, any matrix U∙M where U is unimodular will also be a valid mapping for T. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the quotient group of tempered intervals.
+Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix M is said to be a mapping matrix for a temperament T if and only if the right nullspace of M consists of the kernel of T, M is of full row rank, and the rows of M generate a subgroup of the dual group of vals which is [[Saturation|saturated]] (or [[defactored]]). There is generally not a unique matrix M satisfying this definition for arbitrary temperament T, as for any M which is a valid mapping for T, any matrix U∙M where U is unimodular will also be a valid mapping for T. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the quotient group of tempered intervals.
 The integer column span of any mapping matrix is the quotient group of T-tempered intervals, also known as the quotient group of '''tempered monzos''' [[Tmonzos_and_Tvals|tmonzos]] for T. The integer row span of any mapping matrix for a temperament T is the subgroup of vals that all support T. Note also that this means that if T is of rank r, then any rank-r matrix in which the rows span the subgroup of vals supporting T will be a valid mapping for T.